Cohen Generic Structures with Functions
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929375170002944 |
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| author | Ackerman, Nathanael Golshani, Mohammad Mirabi, Mostafa |
| author_facet | Ackerman, Nathanael Golshani, Mohammad Mirabi, Mostafa |
| contents | Suppose $\mathscr{L}^-\subseteq \mathscr{L}$ are languages where $\mathscr{L} \setminus\mathscr{L}^-$ is relational. Additionally, let $\mathbf{K}$ be a strong $\textrm{Fraïssé}$ class in $\mathscr{L}$. We consider the partial ordering, under substructure, of those elements in $\mathbf{K}$ whose reduct to $\mathscr{L}^-$ are substructures of a fixed $\mathscr{L}^-$-structure $\mathcal{M}^-$. In this paper, we establish that, under general conditions, this partial order satisfies the $|\mathcal{M}^-|$-chain condition.
Furthermore, under these conditions, we demonstrate that any generic for such a partial order satisfies the theory of the \Fraisse\ limit of $\mathbf{K}$, provided $\mathcal{M}^-$ satisfies the theory of$\textrm{Fraïssé}$ limit of its age.
We also provide general conditions that guarantee all such generics to be rigid, as well as conditions ensuring that these generics possess large automorphism groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11582 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cohen Generic Structures with Functions Ackerman, Nathanael Golshani, Mohammad Mirabi, Mostafa Logic 03C55, 03E35 Suppose $\mathscr{L}^-\subseteq \mathscr{L}$ are languages where $\mathscr{L} \setminus\mathscr{L}^-$ is relational. Additionally, let $\mathbf{K}$ be a strong $\textrm{Fraïssé}$ class in $\mathscr{L}$. We consider the partial ordering, under substructure, of those elements in $\mathbf{K}$ whose reduct to $\mathscr{L}^-$ are substructures of a fixed $\mathscr{L}^-$-structure $\mathcal{M}^-$. In this paper, we establish that, under general conditions, this partial order satisfies the $|\mathcal{M}^-|$-chain condition. Furthermore, under these conditions, we demonstrate that any generic for such a partial order satisfies the theory of the \Fraisse\ limit of $\mathbf{K}$, provided $\mathcal{M}^-$ satisfies the theory of$\textrm{Fraïssé}$ limit of its age. We also provide general conditions that guarantee all such generics to be rigid, as well as conditions ensuring that these generics possess large automorphism groups. |
| title | Cohen Generic Structures with Functions |
| topic | Logic 03C55, 03E35 |
| url | https://arxiv.org/abs/2310.11582 |